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Answer:
Step-by-step explanation:
Hello!
In the study population, there is an incidence of two diseases, a serious disease (D) and a minor disease (M).
There is a diagnostic test for the serious disease that successfully detects the disease (positive) in 95% of all persons infected (D). Symbolized P(+/D)=0.95
This test incorrectly diagnoses 4% (positive) of all healthy people (H) as having a serious disease. Symbolized P(+/H)= 0.04
And it incorrectly diagnoses 12% of all people having another minor disease as having a serious disease. Symbolized P(+/M)= 0.12
It is known in the population that:
2% of the population has a serious disease. P(D)= 0.02
90% of the population is healthy. P(H)= 0.90
8% of the population has a minor disease. P(M)= 0.08
I've attached a contingency table with the symbolized probabilities you need to calculate for each category.
Remember: Given the dependent events A and B a conditional probability of this events is defined as P(A/B)= P(A∩B)/P(B)
⇒ Then you can clear the probability of the intersection of both events as:
P(A∩B)= P(A/B)*P(B)
You can apply this to the asked probabilities:
P(+∩D)= P(+/D)*P(D)= 0.95*0.02= 0.019
P(+∩M)= P(+/M)*P(M)= 0.12*0.08= 0.0096
P(+∩H)= P(+/H)*P(H)= 0.04*0.90= 0.036
P(+) = P(+∩D) + P(+∩M) + P(+∩H)= 0.019 + 0.0096 + 0.036= 0.0646
P(-) = 1 - P(+) = 1 - 0.0646= 0.9354
P(-∩D)= P(D) - P(+∩D)= 0.02 - 0.019= 0.001
P(-∩M)= P(M) - P(+∩M)= 0.08 - 0.0096=0.0704
P(-∩H)= P(H) - P(+∩H)= 0.90 - 0.036= 0.864
Tree diagram and completed table attached below.
I hope it helps!
